A055665 Number of Eisenstein-Jacobi primes of successive norms (indexed by A055664).

6, 6, 12, 12, 12, 6, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 6, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 6, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 6, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12

Offset

1,1

Comments

These are the primes in the ring of integers a+b*omega, a and b rational integers, omega = (1+sqrt(-3))/2.

References

R. K. Guy, Unsolved Problems in Number Theory, A16.

L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. VI.

Links

Antti Karttunen, Table of n, a(n) for n = 1..1000 (computed from the b-file of A055666 with the formula of Franklin T. Adams-Watters)

Formula

a(n) = 6 * A055666(n) - Franklin T. Adams-Watters, May 05 2006

Example

There are 6 Eisenstein-Jacobi primes of norm 3, omega-omega^2 times one of the 6 units [ +-1, +-omega, +-omega^2 ] but only one up to equivalence.

Mathematica

norms = Join[{3}, Select[Range[1000], (PrimeQ[#] && Mod[#, 6] == 1) || (PrimeQ[Sqrt[#]] && Mod[Sqrt[#], 3] == 2) &]]; r[n_] := Reduce[n == a^2 - a*b + b^2, {a, b}, Integers] // Length; A055665 = r /@ norms (* Jean-François Alcover, Oct 24 2013 *)

Crossrefs

Cf. A055664-A055668, A055025-A055029. See A004016 and A035019 for theta series of Eisenstein (or hexagonal) lattice.

Sequence in context: A220435 A340512 A338992 * A168328 A176569 A315778

Adjacent sequences: A055662 A055663 A055664 * A055666 A055667 A055668

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, Jun 09 2000

Extensions

More terms from Franklin T. Adams-Watters, May 05 2006

Status

approved